Ratios and Proportional Relationships

Transcription

1 Ratios and Proportional Relationships Standards Entry Points Access Skills 6 Page 86 Pages Pages Page 89 Page 90 MATHEMATICS 85

2 CONTENT AREA Mathematics DOMAIN Ratios and Proportional Relationships Cluster Understand concepts and use reasoning to solve Standards as written 6.RP.A.1 Grade 6 GRADE 6 Mathematics Ratios and Proportional Relationships Understand the concept of a including the distinctions between part:part and part:whole and the value of a ; part/part and part/whole. Use language to describe a relationship between two quantities. For example: The of wings to beaks in the bird house at the zoo was 2:1, because for every two wings there was one beak; For every vote candidate A received, candidate C received nearly three votes, meaning that candidate C received three out of every four votes or ¾ of all votes. 6.RP.A.2 Understand the concept of a unit rate a/b associated with a a:b with b 0, and use rate language in the context of a relationship, including the use of units. 6.RP.A.3 6.RP.A.3a 6.RP.A.3b For example: This recipe has a of three cups of flour to four cups of sugar, so there is ¾ cup of flour for each cup of sugar; We paid $75 for 15 hamburgers, which is a rate of five dollars per hamburger. 1 Use and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent s, tape diagrams, double number line diagrams, or equations. Make tables of equivalent s relating quantities with whole-number measurements. Find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare s. Solve unit rate problems, including those involving unit pricing, and constant speed. For example, if it took seven hours to mow four lawns, then, at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? 6.RP.A.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means times the quantity); solve problems involving finding the whole, given a part and the percent. 6.RP.A.3d 6.RP.A.3e Use reasoning to convert measurement units within and between measurement systems; manipulate and transform units appropriately when multiplying or dividing quantities. For example, Malik is making a recipe, but he cannot find his measuring cups! He has, however, found a tablespoon. His cookbook says that 1 cup = 16 tablespoons. Explain how he could use the tablespoon to measure out the following ingredients: two cups of flour, ½ cup sunflower seed, and 1¼ cup of oatmeal. 2 Solve problems that relate the mass of an object to its volume. 1 Expectations for unit rates in this grade are limited to non-complex fractions. 2 Example is from the Illustrative Mathematics Project: MATHEMATICS 86

3 ENTRY POINTS and ACCESS SKILLS for Ratios and Proportional Relationships Standards in Grade 6 Less Complex More Complex Understand concepts and use reasoning to solve ACCESS SKILLS ENTRY POINTS The student will: The student will: The student will: The student will: Respond to materials that Track materials that Shift focus from materials that Grasp materials that demonstrate s and Use two hands to hold materials that Release materials used to Orient materials used to Locate objects partially hidden or out of sight (e.g., remove barrier) to expose a Use one object to act on another used to demonstrate s Create s among objects sorted by attribute (e.g., of circles to squares is 5:3) Identify a part-topart relationship in a real-life situation using a proportion (e.g. 8 boys to 3 girls or 8:3 boys to girls) Express percents using drawings or technology (e.g., 50% = ½ of a circle) Express percents as fraction equivalents (e.g., 75% = ) Convert simple measurement units (e.g., feet to yards) Determine if given points graphed on a coordinate plane lie on a line that passes through the origin (and thus represent a relationship) Express part-to-part s in mathematical or real-life situations (e.g., 6 blue marbles to 8 green marbles is a of 6:8) Express the mathematical relationship of two known quantities as a (e.g. the of beaks to wings is 1:2 and the of claws to beaks is 8:1) Plot equivalent s as ordered pairs, in the first quadrant of a coordinate plane (e.g. (2, 4), (3, 6), etc.) Convert measurement units from different measurement systems (e.g. kilometers to miles) Identify the factor used to obtain equivalent fractions (e.g., because 1 and ) Express part-towhole s in mathematical or real-life situations (e.g. given 6 blue marbles and 8 green marbles, of the marbles are blue) Represent a unit rate between two given quantities (e.g., if 16 cupcakes are split between 8 children then unit rate is 2 cupcakes per child; 5 apples cost $2.00 so the unit rate is $0.40 per apple) Calculate unit rates in real-life problems to make comparisons (e.g., compare prices from Home Depot and Lowes by calculating unit rates) Convert rates by manipulating measurement units (e.g., miles per hour to feet per second) MATHEMATICS 87

4 ENTRY POINTS and ACCESS SKILLS for Ratios and Proportional Relationships Standards in Grade 6 Less Complex More Complex Understand concepts and use reasoning to solve ACCESS SKILLS ENTRY POINTS The student will: The student will: The student will: The student will: Turn on/off technology used to (e.g., turn on voicegenerating device to describe a relationship using to/for every language) Imitate action to create Initiate cause-andeffect response (e.g., turn on technology tool to activate computer program) Sustain and relationship activity through response Gain attention during a activity Make a request during activity Choose from an array of two in adding and/or subtracting activity (e.g., choose materials to be distributed in a and relationship activity) Attend visually, aurally, or tactilely to materials that Calculate unit rates in real-life problems from a table showing a relationship (e.g., determine the price of one item from a table showing various numbers and total costs of the item) Note: A relationship is one in which the quotient of every y value to its corresponding x value is a constant. Graphically, at this level, the relation is a line in the first quadrant that passes through the origin. MATHEMATICS 88

5 CONTENT AREA Mathematics DOMAIN Ratios and Proportional Relationships Cluster Analyze and use them to solve realworld and mathematical 7.RP.A.1 7.RP.A.2 7.RP.A.2a 7.RP.A.2b 7.RP.A.2c 7.RP.A.2d 7.RP.A.3 Standards as written Grade 7 Compute unit rates associated with s of fractions, including s of lengths, areas, and other quantities measured in like or different units. For example, if a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction ½ ¼ miles per hour, equivalently 2 miles per hour. Recognize and represent between quantities. Decide whether two quantities are in a relationship, e.g., by testing for equivalent s in a table, or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. Identify the constant of ity (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of. Represent by equations. GRADE 7 Mathematics Ratios and Proportional Relationships For example, if total cost t is to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. Explain what a point (x, y) on the graph of a relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Use to solve multi-step, rate, and percent For example, simple interest, tax, price increases and discounts, gratuities and commissions, fees, percent increase and decrease, percent error. MATHEMATICS 89

6 ENTRY POINTS for Ratios and Proportional Relationships Standards in Grade 7 Less Complex More Complex Analyze and use them to solve realworld and mathematical The student will: The student will: The student will: Calculate a unit rate in the context of a described relationship Complete a table of a relationship by filling in missing values (e.g., a recipe that calls for 4 Compute unit rates in terms cups of flour and 2 cups of of distance and time (e.g., If sugar has a of 2 cups of Jaycee traveled 20 miles in flour to 1 cup of sugar) 30 minutes, then she Find equivalent s for a given in a real-life traveled at a rate of 40 miles per hour) situation (e.g., if the of Solve percentage proportions girls to boys is 2:3, then for where one missing quantity 40 girls there are 60 boys, is represented by a variable and for 30 girls there are 45 (e.g., boys) Create a proportion table Calculate the percentage in a given a (e.g., given the percent problem (e.g., what 1:3, create a table of percent of 60 is 45?) equivalent s such as 2:6, Calculate the base in a 3:9, 6:18, 12:36, etc.) percent problem (e.g., 9 is Identify the unit rate from a 75% of what number?) table, graph, equation, or a Calculate the unknown description amount in a percent problem Determine if values given in (e.g., what number is 50% a table represent a of 8?) relationship Solve one-step equations (e.g., do the values (3, 5), using multiplication (e.g., (6, 16), (15, 25) and (25, 75) 3x 45 or 4x 36) in table form represent a Determine the amount of tax proportion?) charged on an item given the Determine the amount of tax item cost in whole dollars charged on an item given the and the tax rate (e.g., the item cost and the tax rate cost of an item is $10 and (e.g., the cost of an item is the tax rate is 4%, the $18.99 and the tax rate is amount of tax charged is 5%, the amount of tax $0.40) charged is $0.95) See entry points for earlier grades in this or a related cluster that are challenging and use ageappropriate materials Solve proportions where one missing quantity is represented by a variable (e.g., ) Solve percentage decrease or increase problems (e.g., a dress that originally cost $50 was on sale for 60% off; a train fare of $14 was increased by 20%) Compute unit rates in a variety of contexts (e.g., miles per hour, cents per stick of gum, minutes per commercial) Solve equations that represent in real life (e.g., use drt to find time if distance is 120 miles and the rate is 60 miles per hour) Solve mixed single-step percent problems using (e.g., what is 45% of 80?; 35 percent of what is 7?; 60 is what percent of 200?) Determine the total cost of an item given the item cost and the tax rate (e.g., the cost of an item is $11.95 and the tax rate is 6%, the total cost of the item is $12.67) Determine the tax rate given the pre-tax and the total costs (e.g., the cost of an item is $ and the total cost with tax is $628.95, the tax rate is 5%) MATHEMATICS 90